Apply the function g to the input 10

Oct 13, 2025 by windailys.com 76 views

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Introduction

In mathematics, a function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). When we have two functions, we can combine them to create a new function, known as the composition of functions. In this article, we will explore the concept of composition of functions and use it to solve a problem involving two given functions.

What is Composition of Functions?

Composition of functions is a way of combining two or more functions to create a new function. It is denoted by the symbol ∘ (circle) and is read as "f composed with g". The composition of functions is defined as:

(f ∘ g)(x) = f(g(x))

In other words, we first apply the function g to the input x, and then apply the function f to the result.

Example: Composition of Two Functions

Let's consider two functions:

f(x) = x^2 + 1 g(x) = x - 4

We want to find the value of (f ∘ g)(10). To do this, we need to follow the order of operations:

Apply the function g to the input 10: g(10) = 10 - 4 = 6
Apply the function f to the result: f(g(10)) = f(6) = 6^2 + 1 = 37

Therefore, the value of (f ∘ g)(10) is 37.

Step-by-Step Solution

To solve this problem, we can follow these steps:

Apply the function g to the input 10: g(10) = 10 - 4 = 6
Apply the function f to the result: f(g(10)) = f(6) = 6^2 + 1 = 37

Conclusion

In this article, we have explored the concept of composition of functions and used it to solve a problem involving two given functions. We have seen that the composition of functions is a powerful tool for combining functions and creating new functions. By following the order of operations and applying the functions in the correct order, we can find the value of the composition of functions.

Key Takeaways

Composition of functions is a way of combining two or more functions to create a new function.
The composition of functions is denoted by the symbol ∘ (circle) and is read as "f composed with g".
To find the value of the composition of functions, we need to follow the order of operations and apply the functions in the correct order.

Practice Problems

Find the value of (f ∘ g)(5) given f(x) = x^2 + 1 and g(x) = x - 2.
Find the value of (f ∘ g)(10) given f(x) = x^2 + 2 and g(x) = x - 3.

Solutions

(f ∘ g)(5) = f(g(5)) = f(3) = 3^2 + 1 = 10
(f ∘ g)(10) = f(g(10)) = f(6) = 6^2 + 2 = 38

Final Thoughts

Composition of functions is a fundamental concept in mathematics that allows us to combine functions and create new functions. By following the order of operations and applying the functions in the correct order, we can find the value of the composition of functions. With practice and experience, we can become proficient in using composition of functions to solve a wide range of problems.

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Introduction

In our previous article, we explored the concept of composition of functions and used it to solve a problem involving two given functions. In this article, we will answer some frequently asked questions about composition of functions.

Q: What is the difference between function composition and function multiplication?

A: Function composition and function multiplication are two different mathematical operations. Function composition is a way of combining two or more functions to create a new function, while function multiplication is a way of multiplying two or more functions together. For example, if we have two functions f(x) = x^2 + 1 and g(x) = x - 4, we can compose them to get (f ∘ g)(x) = f(g(x)) = (x - 4)^2 + 1, but we cannot multiply them together.

Q: How do I know which function to apply first in a composition of functions?

A: When composing two functions, we need to apply the inner function first and then the outer function. This is because the output of the inner function becomes the input of the outer function. For example, if we have two functions f(x) = x^2 + 1 and g(x) = x - 4, we need to apply g(x) first and then f(x) to get (f ∘ g)(x) = f(g(x)) = (x - 4)^2 + 1.

Q: Can I compose more than two functions together?

A: Yes, we can compose more than two functions together. For example, if we have three functions f(x) = x^2 + 1, g(x) = x - 4, and h(x) = x + 2, we can compose them to get (f ∘ g ∘ h)(x) = f(g(h(x))) = f(g(x + 2)) = f((x + 2) - 4) = f(x - 2) = (x - 2)^2 + 1.

Q: How do I evaluate a composition of functions?

A: To evaluate a composition of functions, we need to follow the order of operations and apply the functions in the correct order. For example, if we have two functions f(x) = x^2 + 1 and g(x) = x - 4, and we want to find the value of (f ∘ g)(10), we need to apply g(x) first and then f(x) to get (f ∘ g)(10) = f(g(10)) = f(6) = 6^2 + 1 = 37.

Q: Can I use composition of functions to solve real-world problems?

A: Yes, composition of functions can be used to solve real-world problems. For example, if we have a function that represents the cost of producing a certain number of items, and another function that represents the number of items sold, we can compose them to get a function that represents the total revenue.

Q: What are some common applications of composition of functions?

A: Composition of functions has many applications in mathematics, science, and engineering. Some common applications include:

Modeling population growth and decline
Analyzing the behavior of complex systems
Solving optimization problems
Evaluating the performance of algorithms

Conclusion

In this article, we have answered some frequently asked questions about composition of functions. We have seen that composition of functions is a powerful tool for combining functions and creating new functions, and that it has many applications in mathematics, science, and engineering.

Key Takeaways

Composition of functions is a way of combining two or more functions to create a new function.
The composition of functions is denoted by the symbol ∘ (circle) and is read as "f composed with g".
To evaluate a composition of functions, we need to follow the order of operations and apply the functions in the correct order.
Composition of functions has many applications in mathematics, science, and engineering.

Practice Problems

Find the value of (f ∘ g)(5) given f(x) = x^2 + 1 and g(x) = x - 2.
Find the value of (f ∘ g)(10) given f(x) = x^2 + 2 and g(x) = x - 3.
Find the value of (f ∘ g ∘ h)(x) given f(x) = x^2 + 1, g(x) = x - 4, and h(x) = x + 2.

Solutions

(f ∘ g)(5) = f(g(5)) = f(3) = 3^2 + 1 = 10
(f ∘ g)(10) = f(g(10)) = f(6) = 6^2 + 2 = 38
(f ∘ g ∘ h)(x) = f(g(h(x))) = f(g(x + 2)) = f((x + 2) - 4) = f(x - 2) = (x - 2)^2 + 1